MAT444 Aug 07

MAT444 Aug 07 - Qualifying Examination in Linear Algebra...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Qualifying Examination in Linear Algebra and Field Theory August, 2007 Answer as many questions as you can, but be sure to answer sufficiently many questions to comprise 90 points, with at least 30 points from each section. Write your answer to each problem on a separate sheet, placing the problem number in the upper right corner. Be sure to justify all your answers. Do not cite any result that reduces a proof to a triviality. Field Theory 1. [5,7,5] Let K be a field and let be an element of an algebraic closure of K . a. Let f ( X ) K [ X ] be the monic irreducible polynomial of over K and suppose deg f = n . Let L denote the splitting field of f ( X ) over K . Find an upper bound for [ L : K ] in terms of n . Prove that your answer is correct. b. Suppose [ K ( ) : K ] is odd. Prove that K ( 2 ) = K ( ). c. Let E denote the splitting field of X 5- 1 over Q . Find a primitive element for E , i.e., find a complex number so that E = Q ( ). Prove that your answer is correct....
View Full Document

Page1 / 2

MAT444 Aug 07 - Qualifying Examination in Linear Algebra...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online